Method for measuring the location of an object by phase detection

ABSTRACT

A method for measuring the location of an object in an observed space by means of a fixed observation system connected to a processing unit for generation of an image comprising a pixel matrix, the object being provided with a test marker. The test marker comprises a periodic pattern in two dimensions and a digital processing of the image of the test marker is carried out to produce an image comprising a first grating and an image comprising a second grating which are analyzed digitally to calculate the position of the test marker within the matrix of pixels.

The present invention relates to a method for measuring the location ofan object by phase detection.

More particularly, the invention concerns a method for measuring thelocation of an object observed by a fixed observation system connectedto a processing unit, in order to generate an image composed of a matrixof pixels.

In order to locate the object precisely, the latter is provided in amanner which is known per se with a test marker having two periodicgratings, the representation of which in the pixel matrix is formed bytwo periodic gratings intended, after conversion to the frequencydomain, to constitute two bidirectional phase references which can beprocessed by extracting the phase information by means of a frequencyanalysis function such as Morlet wavelet transforms. The phaseinformation detected in this way is then combined in order to determinethe cartesian coordinates of the reference point of the test marker, aswell as the orientation of the test marker with respect to theobservation system. Application of this measurement method to the imageof a suitable test marker, obtained by a standard sensor, allows highresolution in the location of the reference point of the test marker.Such a measurement method is described in an article in the journal“IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT” volume 49, number4, pages 867 to 872.

This measurement method consist basically in using a test marker formedby a first grating comprising a plurality of parallel and regularlyspaced first strips, and by a second grating comprising a plurality ofparallel and also regularly spaced second strips. These first and secondgratings are furthermore arranged so that the first strips aresubstantially perpendicular to the second strips, although the first andsecond gratings are physically separated by a certain distance from oneanother.

The image of this test marker in the pixel matrix obtained by theobservation system, or more specifically the image of the two gratings,is then processed by the processing unit in order basically to carry outthe following operations for each grating:

-   -   locating and extracting the image of the grating from the entire        matrix of pixels,    -   calculating the pixel frequency of this grating along a first        alignment of pixels which intersects all the strips of this        grating,    -   using the pixel frequency of this grating in order to define an        analysis function which is applied to this grating along the        first alignment of pixels,    -   extracting the phase and the modulus which are associated with        this grating, by correlation with the analysis function, in        order to calculate the cartesian position of the middle of at        least one strip of the grating in the direction of the first        alignment of pixels,    -   successively extracting the phase and the modulus which are        associated with this grating, by correlation with the analysis        function along a plurality of pixel alignments which are        parallel to the first alignment of pixels, in order to        independently determine the cartesian position of each middle of        said at least one strip in the direction of each corresponding        alignment of pixels,    -   for each grating, calculating a median line passing        substantially through all the middles of said corresponding at        least one strip, the median line of the first grating being        perpendicular to the median line of the second grating,    -   calculating the cartesian position of the point of intersection        between the two median lines, and    -   calculating the angle defined by the median strip of the first        grating and a predetermined alignment of pixels.

An example of a known device for carrying out the method as describedabove is schematically represented in FIG. 1.

This device comprises an observation system 1 comprising a matricialimage sensor such as a CCD camera 2, and a lens 3 for forming the imageof the observed scene on the matricial image sensor 2. This matricialimage sensor is connected to a processing unit 4 intended to make itpossible to analyze the phase of the image formed by a pixel matrixobtained using the matricial image sensor 2. This processing unit 4 isalso designed to carry out logical and arithmetic operations on therecorded images coming from the matricial image sensor. In order topermit position measurements of an object 5 moving in the fixedobservation field of the sensor 2, a test marker 6 is fixed on thismobile object 5. The test marker 6 comprises a first grating P1 formedby N1 parallel and regularly spaced strips, and a second grating P2formed by N2 parallel and regularly spaced strips. These two gratingsP1, P2 are physically separated from each other, and the N1 strips aresubstantially perpendicular to the N2 strips of the second grating P2.

The gratings P1 and P2 are, for example, etched by photolithography on aglass mask, the latter being illuminated by a lighting device making itpossible to obtain a matrix of pixels, representing the images of thegratings P1 and P2, from the matricial image sensor.

After the various operations of processing the recorded image of thistest marker 6 by means of the processing unit 4, these variousprocessing operations being described in more detail in the rest of thedescription, a cartesian representation of the test marker 6 is obtainedas can be seen in FIG. 2.

In the example in question, the processing unit 4 therefore makes itpossible to calculate the equation of a line D2 of the grating P2 andthe equation of a median line D1 of the grating P1, these median linesD1 and D2 being respectively defined by all the middles of the centralstrip of each grating P1 and P2 in the assembly in question. Theposition of the test marker 6, and therefore of the object 5, is givenby the cartesian coordinates Δx, Δy of the point of intersection P ofthe two median lines D1 and D2.

The orientation of the object 5 is in turn defined by the angle θformed, for example, by the median line D1—selected as a reference—ofthe grating P1 with one of the cartesian reference-coordinate axes x, yprovided, for example, by the matrix of pixels constituting the image.

With this type of measurement method, and after recording and analysisof two consecutive images of the test marker 6, it is possible to detectdisplacements of the object 5 with a precision of the order of 1.10⁻²pixel.

But the use of a test marker that has two periodic gratings physicallyseparated from each other, as can be seen in FIG. 2, means that thepoint of intersection P of the two median lines D1 and D2 lies insidethe second grating P2 but at a relatively large distance from the firstgrating P1. In the event that there is the slightest error in thecalculation of the slope of the reconstructed median line D1, therefore,it can be seen that this error will automatically be passed on to theposition of the point P along the median line D2. This positioning errorof the point P along the median line D2 will be commensurately largerwhen the grating P1 is further away from the grating P2.

It is, in particular, an object of the invention to overcome theaforementioned drawbacks.

To this end, according to the invention, the measurement method of thetype in question is essentially characterized in that it comprises thefollowing steps:

-   -   a test marker is used comprising at least one two-dimensional        periodic pattern formed by a plurality of substantially        point-like elements arranged in parallel rows and parallel        columns, which are substantially perpendicular to the rows, the        point-like elements being regularly spaced along the rows and        the columns,    -   a first image of the pattern is recorded, and digital processing        of the first image of the pattern is carried out in order to        generate, from said pattern, an image containing a first grating        comprising a plurality of regularly spaced parallel first strips        and an image containing a second grating comprising a plurality        of regularly spaced parallel second strips, the second strips        being substantially perpendicular to the first strips, and for        each of the first and second gratings,    -   the pixel frequency of this grating is calculated along a first        alignment of pixels which intersects all the strips of this        grating,    -   the pixel frequency of this grating is used to define an        analysis function which is applied to this grating along the        first alignment of pixels,    -   the phase and the modulus which are associated with this grating        are extracted by correlation with the analysis function in order        to calculate the cartesian position of the middle of at least        one strip of the grating in the direction of the first alignment        of pixels,    -   the phase and the modulus which are associated with this grating        are successively extracted by correlation with the analysis        function along a plurality of pixel alignments which are        parallel to the first alignment of pixels, each alignment of        pixels intersecting all the strips of this grating, in order to        independently determine the cartesian position of each middle of        said at least one strip in the direction of each corresponding        alignment of pixels,    -   a median line passing substantially through all the middles of        said at least one strip is calculated for each grating, the        median line of the first grating being perpendicular to the        median line of the second grating,    -   the cartesian position of the point of intersection between the        two median lines is calculated, and    -   the angle defined by the median line of the first grating and a        predetermined alignment of pixels is calculated.

In preferred embodiments of the invention, one and/or other of thefollowing arrangements may optionally be employed as well:

-   -   a second image of said at least one periodic pattern is recorded        after a displacement of the object in the space observed by the        fixed observation system, and the cartesian position of the        point of intersection of the two median lines of the first and        second gratings as obtained from the second recorded image is        calculated in order to calculate the displacement of the object;    -   the digital processing of the first image of said at least one        periodic pattern comprises the following steps:        -   a forward Fourier transform is applied to the image of the            pattern in order to obtain the Fourier spectrum of the image            of said periodic pattern,        -   based on the Fourier spectrum, two independent filtering            operations are carried out in order to obtain, on the one            hand, a first filtered Fourier spectrum associated with the            direction of the columns of the periodic pattern and, on the            other hand, a second filtered Fourier spectrum associated            with the direction of the rows of the periodic pattern, and        -   an inverse Fourier transform is applied to each of the first            and second filtered Fourier spectra in order to obtain the            image of the first grating and the image of the second            grating;    -   the test marker comprises a matrix of identical periodic        patterns arranged in parallel rows and parallel columns, which        are substantially perpendicular to the rows, the periodic        patterns being regularly spaced along the rows and the columns,        and each periodic pattern being associated with a positioning        element for locating the periodic pattern which is associated        with it inside the matrix of periodic patterns;    -   each positioning element comprises a row number index and a        column number index for making it possible to locate the pattern        which is associated with it inside the matrix of periodic        patterns;    -   the image of each row number index and column number index in        the matrix of pixels is in the form of a barcode which is read        by the processing unit;    -   the fixed observation system comprises a first and a second        matricial image sensor which are contained substantially in a        plane perpendicular to a plane defined by the two dimensions of        the periodic pattern of the test marker, the first and second        image sensors having sighting axes each of which delimits a        predetermined angle with the axis perpendicular to the plane,        and        -   an image of said at least one periodic pattern is recorded            by each sensor,        -   the first cartesian position of the point of intersection as            obtained from the first sensor is calculated,        -   the second cartesian position of the point of intersection            as obtained from the second sensor is calculated, and        -   based on the first and second cartesian positions of the            point of intersection and the predetermined angles, the            position of the point of intersection is calculated along a            direction parallel to the plane defined by the two            dimensions of said at least one periodic pattern and a            direction perpendicular to the plane defined by the two            dimensions of said at least one periodic pattern;    -   the fixed observation system comprises a first matricial image        sensor that has a sighting axis perpendicular to the plane        defined by the two dimensions of the periodic pattern of the        test marker and a second matricial image sensor that has a        sighting axis parallel to the plane defined by the two        dimensions of the periodic pattern of the test marker, a        light-beam splitting object being furthermore interposed between        the periodic pattern and the first and second sensors, and        -   an image of said at least one periodic pattern is recorded            by each sensor,        -   the cartesian position of the point of intersection in a            plane parallel to the plane defined by the two dimensions of            the periodic pattern is calculated from the image obtained            by the first sensor, and        -   the cartesian position of the point of intersection in a            plane perpendicular to the plane defined by the two            dimensions of the periodic pattern is calculated from the            image obtained by the second sensor;    -   the frequency of the periodic pattern, as calculated by the        processing unit, is compared with the real frequency of the        periodic pattern in order to determine the position of the point        of intersection in a direction perpendicular to the plane        defined by the two dimensions of said at least one periodic        pattern, as a function of the magnification index of the fixed        observation system.

Other characteristics and advantages of the invention will becomeapparent from the following description of one of its embodiments, whichis given by way of a nonlimiting example, with reference to the appendeddrawings.

IN THE DRAWINGS:

FIG. 1 represents the device for carrying out the aforementioned methodaccording to the prior art,

FIG. 2 represents an example of a strip grating according to the priorart for the position calculation,

FIG. 3 represents a measurement device for carrying out the methodaccording to the invention,

FIG. 4 represents a test marker according to the invention forfacilitating a position calculation,

FIG. 5 represents an image of the test marker according to theinvention, obtained using the observation system of the device,

FIG. 6 represents an enlargement of a portion of the image of the testmarker in FIG. 5,

FIG. 7 represents the Fourier spectrum of the image of the test markeraccording to the invention,

FIGS. 8 a and 8 b represent a reconstruction of the Fourier spectrum,respectively in the direction of the columns of the test marker and inthe direction of the rows of the test marker,

FIGS. 9 a and 9 b are views of the pixel-based spatial representationsof two gratings, obtained by the frequency processing of the image ofthe test marker,

FIGS. 10 a and 10 b represent regions of interest in the gratings ofFIGS. 9 a and 9 b, for facilitating the position calculation,

FIG. 11 represents the intensity of the signal emitted by the grating inFIG. 10 b along a column,

FIG. 12 is a view of the Fourier spectrum of the intensity of the signalas represented in FIG. 11,

FIG. 13 represents the modulus of the wavelet transform along the columnC_(c) as represented in FIG. 10 b,

FIG. 14 represents the phase of the wavelet transform along the columnC_(c) as represented in FIG. 10 b,

FIG. 15 represents the product of the derivative of the modulus asrepresented in FIG. 13 multiplied by the phase as represented in FIG. 14(the peaks defining the ends of the strip grating along the column C_(c)represented in FIG. 10 b);

FIG. 16 represents the superposition of the developed phase and theintensity along the column C_(c) in FIG. 10 b,

FIGS. 17, 18 and 19 represent the images of the strip gratings asreconstructed by the digital processing, as well as the secant line ascalculated from each of the strip gratings and their point ofintersection that represents the position of the mobile object withrespect to the fixed reference coordinates formed by the frame of thepixels of the recorded image,

FIG. 20 represents an alternative embodiment of the test markeraccording to the invention,

FIGS. 21 and 22 represent positioning elements intended to be formed onthe test marker represented in FIG. 20,

FIG. 23 represents an alternative embodiment of the device for carryingout the method according to the invention,

FIG. 24 represents another alternative embodiment of the device forcarrying out the method according to the invention, and

FIG. 25 represents yet another alternative embodiment of the device forcarrying out the method.

In the various figures, references that are the same denote identical orsimilar elements.

FIG. 3 represents an example of a measurement device needed for carryingout the method according to the invention. This device comprises amatricial image sensor such as a CCD camera 2, a microscope objective 3and a matching tube 7 that connects the sensor 2 to the microscopeobjective 3 in order to form the observation system 1 of said device.The device may, of course, simply comprise an imaging lens and amatricial image sensor. This observation system 1 is intended to remainimmobile. An object 5 is placed in the field of view of the sensor 2 andthis object 5 is provided with a test marker 8 fixed on the support ormore specifically, in the example in question, on a back-lighting table13 itself fixed on the object 5. This object 5 is intended to move in atwo-dimensional space defined by the plane [xoy]. Furthermore, thesensor 2 is also arranged so that its viewing axis 2 a is substantiallyperpendicular to the plane [xoy]. In this embodiment, the test marker 8as represented in FIG. 4 comprises a two-dimensional periodic pattern 8a formed by a plurality of point-like elements 9 arranged in parallelrows (of which there are 12 in the example in question) and parallelcolumns (of which there are also 12 in the example in question) whichare perpendicular to the rows. The test marker 8 may, for example, beformed by a glass mask 8 b covered with a layer that is opaque over itsentire surface and in which the transparent point-like elements 9 areobtained by photolithography, so that the surface of the test marker 8is opaque except at the point-like elements 9. The number of rows andcolumns of the test marker 8 may of course vary significantly accordingto the type of test marker used, without thereby departing from thescope of the invention.

In order to obtain an image of the test marker by means of the sensor 2,the test marker 8 is arranged above a diffuse lighting table 13 so thatthe point-like elements 9 produce luminous points that are distributedover the dark background of the test marker and can be detected by thematricial image sensor. One variant might consist in providing thepoint-like elements 9 with a different reflectivity than the rest of thetest marker, so that these point-like elements have a differentluminosity than the rest of the test marker, the assembly beingilluminated from below.

The test marker 8 is furthermore arranged so that its periodic pattern 8a is substantially arranged in the reference plane [xoy].

The distance d1 between two rows of the periodic pattern 8 a and thedistance d2 between two columns are constant, while the distance d2 maybe equal to or different than the distance d1.

For example, the point-like elements 9 may be of substantially squareshape with sides that have a length of the order of 5 μm.

Of course, the test marker 8 may also be formed by any support on whichthe point-like elements 9 are arranged, which may also be in the form ofreflective elements that reflect the light from an excitation sourceilluminating the test marker 8 so as to obtain an image of a periodicgrating at the sensor.

Likewise, according to an alternative embodiment, the test marker 8 mayalso be formed by a support on which a plurality of periodicthrough-holes 9 are formed, making it possible to obtain an image of aperiodic grating after illumination by a back-lighting table.

FIG. 5 represents an image of the test marker 8 as represented in FIG.4, this image being taken by a CCD sensor with a matrix of pixelsmeasuring 578 pixel rows by 760 pixel columns.

The first step of the method consists in carrying out preliminarydigital processing of the image of this test marker 8 in order tocomputer-generate two separate images, respectively representing a firstgrating formed by a first series of parallel strips and a second gratingformed by a second series of parallel strips, which are perpendicular tothe first series of strips.

To this end, the processing unit 4 (FIG. 3) of the device is used torecord the image of the test marker 8, which is represented in FIG. 5and is obtained by the CCD sensor.

On the basis of the image of this test marker 8, an enlargement of whichis represented in FIG. 6, frequency processing of this image is firstcarried out in order to change from the spatial domain to a frequencydomain. This frequency processing consists, for example, of a forwardFourier transform in order to obtain the Fourier spectrum of therecorded image of the two-dimensional periodic pattern 8 a of the testmarker 8, as can be seen in FIG. 7. On the basis of this Fourierspectrum, two suitable and independent filtering operations are carriedout in order to obtain, on the one hand, a filtered Fourier spectrumassociated with the direction of the rows of the periodic pattern of thetest marker (FIG. 8 a) and, on the other hand, a filtered Fourierspectrum associated with the direction of the columns of the periodicpattern of the test marker (FIG. 8 b).

An inverse Fourier transform is then applied to each of the filteredFourier spectra as represented in FIGS. 8 a and 8 b in order to obtainthe images of two gratings R1 and R2 (FIGS. 9 a and 9 b) in apixel-based spatial representation, these two gratings R1 and R2 beingrepresentative of the periodic pattern 8 a of the test marker 8.

In the example considered in FIGS. 9 a and 9 b, the grating R1 istherefore formed by 12 mutually parallel and substantially verticalstrips T1, while the grating R2 is formed by 12 likewise mutuallyparallel but substantially horizontal strips T2.

Advantageously, the phase information associated with the rows and thecolumns of the periodic pattern 8 a is preserved by this frequencyprocessing of the recorded image of the test marker 8 as represented inFIG. 6, and the gratings R1 and R2 generated in this way contain all thepositional information already available from the test marker 8, or morespecifically from the recorded digital image of the periodic pattern 8 aof the test marker 8.

Calculation of the location of the test marker 8 in the image thusequates to respectively calculating the location of the grating R1 inthe first generated image and the location of the grating R2 in thesecond generated image.

In order to make it possible to calculate the position of each grating,a calculation which equates to determining the position and theorientation of each grating in its image, a region of interest R10, R20is first defined for each grating R1, R2. This region of interest R10,R20 in each grating R1, R2 is determined by systematically excluding theextreme edges of the strips T1, T2.

Each region of interest R10, R20 comprises sides which are pairwiseparallel to the axes defined by the pixel frame of the sensor, that isto say the row axis and the column axis of the matrix of pixels.

When the orientation of the gratings R1 and R2 makes the regions R10,R20 very narrow, a prior rotation of the recorded image of the testmarker is thus applied so that the regions of interest R10, R20 arelarge enough to ensure accuracy of the measurements.

The subsequent processing operations are only carried out in theseregions of interest R10, R20, which represent the only parts of theimages of the gratings R1 and R2 that can be used for the position andorientation calculation.

FIGS. 10 a and 10 b respectively give the pixel-based images of the tworegions of interest R10, R20.

For each region of interest R10, R20, the pixel coordinates of the upperleft-hand edge of the region of interest as well as its height and itswidth in pixels are also determined with respect to the original imageas represented in FIG. 5. The following are thus obtained in theexamples in question:For R1 X₀=235; Y₀=240; Height=107 pixels and width=205 pixelsFor R2 X₀=205; Y₀=240; Height=170 pixels and width=105 pixels

In the rest of the description, we will determine the position and theorientation of each grating in the original image of the test marker 8as given in FIG. 5.

Given that the various processing operations to be described below areidentical for the gratings R1 and R2, in what follows we will only studythe case of the grating R2 formed by 12 substantially horizontal stripsT2 with reference to the pixel rows of the image.

The pixel-based spatial frequency of the grating R2 is determined firstof all. The spatial frequency of the grating R2 is determined, forexample, by Fourier transformation. The frequency of the imaged stripgrating corresponds to a maximum in the Fourier spectrum.

To this end, a column of pixels C_(c) is considered (FIG. 10.b) alongwhich the intensity of the signal received by the matricial image sensoris determined. On the-basis of the intensity of the signal along thecolumn C_(c) as represented in FIG. 11, the processing unit determinesthe Fourier spectrum of the intensity of the signal along this columnC_(c), as indicated in FIG. 12. After exclusion of the low frequenciesof the image, which correspond to the continuous background, it is thenpossible to extract the spatial frequency f_(o) of the imaged gratingR2.

In order to avoid making an error when determining the frequency of thegrating, it is also possible to use all the a priori knowledge about theimaged grating R2. For instance, knowing the number of substantiallyhorizontal strips of the imaged grating R2, this number of strips beingidentical to the number of rows of elementary elements 9 in the patternof the test marker 8, and knowing the approximate size of the region ofinterest R20 of the grating R2, that is to say its height and its width,it is possible to ascertain approximately the period of the grating inpixels and therefore its frequency f_(o).

An analysis function is then constructed for this same frequency f_(o)of the grating R2, this frequency being determined on the basis of thepixel column C_(c) with reference to the pixel matrix of the matricialimage sensor.

For example, the analysis function may be a Morlet wavelet which makesit possible, by correlation with the grating R2, to extract the phaseand the modulus that are associated with this grating.

The Morlet wavelet at the frequency f_(o) for processing the image is ofthe form:Ψ(y)=exp−(y/Lw)².expj(2πf _(o) y)where Lw defines the width of the wavelet. This parameter Lw can proveto be important because the choice of its value determines thecompromise between the spatial and frequency resolutions. For instance,a short wavelet makes it possible to obtain a good spatial resolution,but the information about the phase is very poor in this case. In theconverse case of a long wavelet, the spatial information is insufficientbut a good resolution is obtained for the phase.

In the case of a discrete signal like that delivered by a matrix imagesensor such as a CCD camera, it is of course necessary to introduce adiscrete form of the wavelet in the form:Ψ(j)=exp−(i/Lw)².expj(2πf _(o) i)where i is an integer value lying between −M and M. The value of M mustin this case be matched to the length of the wavelet, that is to say theparameter Lw, in order to insure a complete representation of thewavelet.

For each position k along a column 1 parallel to the column C_(c) (FIG.10.b), the coefficient W_(k), 1 of the wavelet transform is thus givenby the following expression:$W_{k},{l = {\sum\limits_{i = {- M}}^{+ M}\quad{{I\left( {{k + i},l} \right)}\psi_{i}}}}$where I(k, l) is the intensity of the pixel k in the column l.

Since the purpose of the image processing is to reconstruct the totalphase excursion of the imaged grating R2, which is equal to 2Nπ were Nis equal to the total number N2 of strips of the grating R2, it istherefore necessary to extract the phase of the wavelet transform, whichis itself equal to 2Nπ apart from the noise.

The phase and the modulus are respectively given by the argument and themodulus of the complex number W_(k,l). But since the frequency of thewavelet is fixed at f^(o), which is the pixel frequency of the gratingR2 along the column C_(c), the wavelet transform of the grating R2equates to a convolution between the wavelet and the imaged grating R2in one direction.

After calculation, the processing unit thus makes it possible to extractthe modulus and the phase of the wavelet transform along the columnC_(c). The representations of the modulus and the phase of this wavelettransform along the column C_(c) are respectively given by FIGS. 13 and14.

It is then necessary to determine the edges of the grating R2 along thecolumn C_(c), in order to extract the useful part of the phase of thewavelet transform. Specifically, the purpose of the digital processingis to reconstruct the phase excursion 2Nπ or N=12 in the example inquestion.

The following operation may in particular be used to this end, where thederivative of the modulus is multiplied by the phase of the wavelettransform. More specifically, the following operation may be carriedout:B(i,j)=M′(i,j)×|P(i,j)−π|where M′(i,j) is the derivative of the modulus of the wavelet transformalong the column j, i is the index of the row, and P(i,j) is the phaseof the wavelet transform.

The result of this operation along the column C_(c) is represented inFIG. 15, where the indices ib₁ and ib₂ correspond respectively to theupper and lower edges of the grating R2 along the column C_(c).

The indices ib₁ and ib₂ now being perfectly determined, it is thenpossible to reconstruct the phase excursion of 2Nπ. The processing unitis used to carry out superposition of the phase developed over theentire grating, that is to say between ib₁, and ib₂, and the intensityvariation along the column C_(c), as represented in FIG. 16.

After having reconstructed the phase excursion, the least squares linethat passes through the points of the developed phase is thencalculated. The calculation is limited to a region Z1 (FIG. 16) wherethe phase calculation is optimal for avoiding the errors due to edgeeffects.

This least squares line makes it possible to convert from the discretedomain of the image to a continuous space, this least squares linehaving an equation:J=I.a+bwhere I and J are continuous variables.

It is deduced from this equation that the centre of the N2 strips of thegrating as well as the centre of the bands lying between two strips ofthe grating R2 are solutions of the two equation of the following type:(2k−1).π=Ia+b; for the strips with 1<k<nand 2kπ=Ia+b; for the bands with −1<k<nwhere b corresponds to the cartesian ordinate at the origin and acorresponds to the slope of the least squares line.

The equations may of course be different according to whether theobservation is bright on a dark background or dark on a brightbackground, which will depend on the test marker and the lighting whichis used.

Based on these equations, the subpixel position of the middle of thestrips and the bands of the grating R2 along the column C_(c) are thendetermined.

At this stage of the processing of the imaged grating R2, for example,the middle of the sixth strip of the grating R2 along the column C_(c)may be adopted as a reference point. All of the processing describedabove is then repeated for a plurality of pixel columns which areparallel to the column C_(c) and which pass through all the N2 strips ofthe imaged grating R2. After scanning the imaged grating, a plurality ofmutually independent points are then obtained which represent thecartesian coordinates of the middle of the sixth line of the grating R2along each pixel column. When all the middles of the sixth strip of thegrating R2 have been calculated, it is then sufficient to determine theleast squares line D2 defined by the alignment of these middles, asrepresented in FIG. 17. When the least squares line D2 or median line D2has been determined, the processing of the imaged grating R1 formed byN1 strips is then carried out (FIG. 10 a).

In order to obtain the least squares line D1 or median line D1 passingthrough all the middles of the sixth strip of the imaged grating R1, itis sufficient to resume all the processing operations described abovewhile scanning the grating R1, or more specifically the region ofinterest R10, along a plurality of pixel rows. The line D1 representedin FIG. 18 is then obtained. At this stage of the processing, it is thensufficient to virtually superpose the two images of the gratings R1 andR2, or at least to project the median line D2 onto the imaged gratingR1, for example, as can be seen in FIG. 19, in order to obtain theintersection of the two lines D1 and D2, which gives the measurementpoint P associated with the test marker 8.

For example, the line D1 has an equationy=6.5221×−2148.6 (D1)and the line D2 has an equationy=−0.1524×+230.4403

Using this measurement method, the position of the point P is determinedwith a precision of the order of one 100^(th) of a pixel. Furthermore,the reconstruction of two imaged gratings R1 and R2 from the periodicpattern 8 a of the test marker 8 makes it possible to superpose theimaged gratings R1 and R2 while preserving the positional information ofthe test marker 8, which makes it possible to obtain a point ofintersection P lying inside the two imaged gratings R1 and R2. Thislocation of the point P inside the two imaged gratings makes it possibleto considerably minimize the effect of the least error in thecalculation of the slope of the median lines D1 and D2, which arereconstructed by the processing described above.

In all of the method described above, the test marker 8 comprises asingle periodic pattern 8 a. The presence of a single periodic patternthus makes it possible to measure the subpixel displacement bysuccessively recording two images of the test marker 8. Thus when theobject 5 and therefore the test marker 8 move by a fewer nanometers, asseen above, the position of the illuminated pixels in the image of thetest marker 8 is not modified but their intensity values changeslightly. This is because the light intensity distribution incident onthe pixels of the matricial image sensor changes, giving rise to adifferent recorded image of the test marker, which leads to a differentphase during the digital processing and therefore to measurement of thenew position of the mobile target. The value of the displacement isprovided by the difference between the positions measured before andafter the displacement. In other words, therefore, these variationstogether lead to a significant variation of the phase between the twoimages. This modification of the phase distribution is detected andmeasured by the method described above, which makes it possible tocalculate the new cartesian coordinates of the point of intersection Pof the median lines D1 and D2 for the second recorded image of the testmarker 8. The value of the displacement of the point of intersection P,and therefore of the target object, is thus determined by using theslope of one of the median lines D1 or D2 and the respective cartesiancoordinates of the point of intersection P in the first image and in thesecond recorded image.

When a test marker comprising a single periodic pattern is used,however, the measurement of displacement based on two recorded images islimited by the fact that the entire periodic pattern 8 a of the testmarker 8 must necessarily be contained in the pixel matrix of thematricial image sensor. But in the event that the displacement of thetest marker 8 is too large, at least some of the periodic pattern isliable to leave the field of view of the fixed sensor, which then makesit impossible to determine the position of the point P and the angularorientation of the test marker 8.

According to an alternative embodiment of the invention, which isrepresented in FIG. 20, the test marker 8 is provided with a pluralityof periodic patterns 8 n that are identical to the periodic pattern 8 a.The periodic patterns 8 n are arranged regularly, for exampleperiodically in parallel and regularly spaced rows as well as inparallel columns, which are perpendicular to the rows.

Each periodic pattern is, for example, etched by photolithography. Ascan be seen in FIG. 20, each periodic pattern 8 n has a positioningelement 10 n associated with it, which is designed to store positionalinformation making it possible to locate the periodic pattern associatedwith it actually inside the matrix formed by all the periodic patterns 8n. Each positioning element 10 n includes, for example, a row numberindex and a column number index making it possible to preciselyascertain the position of the periodic pattern 8 n which is associatedwith it inside the matrix of periodic patterns.

Furthermore, the spacing between two adjacent periodic patterns 8 n isphysically known since it was chosen when designing the test marker 8.The displacements can therefore be measured with two degrees ofprecision, that is to say the spacing between two periodic patterns anda subpixel precision actually inside the image of the periodic pattern 8n which is being processed by the processing unit 4. In other words thedisplacements are calculated on the basis of two complementary values,that is to say the spacing between the patterns which are observed inthe recordings before and after displacement, and the position of thepattern which is observed in the pixel matrix of the images that arerecorded before and after displacement. During a first locationmeasurement of the test marker, for example, the processing unit mayprocess the recorded image of a periodic pattern seen in its entirety bythe observation system. This periodic pattern 8 n is localized in thematrix of patterns by its row index i1 and its row index j1. Theprocessing unit can then determine the location point P of this patternby means of the various processing operations described above, thisbeing done for example for the sixth strip of its imaged gratings R1 andR2.

When there is a significant relative displacement of the test marker 8,which corresponds to a displacement in excess of the size of theperiodic pattern processed previously, the field of view of the sensorthen detects another periodic pattern. Owing to its positioning element,this other pattern is localized in the matrix of patterns by its rowindex i2 and its row index j2. The processing unit can then determinethe location point P of this new periodic pattern by taking the sixthstrip of its imaged gratings R1 and R2 as a reference. The displacementof the test marker 8 is therefore deduced from this processing, which inthis example equates to calculating the known spacing between the rowsi1 and i2 and the columns j1 and j2 and the subpixel displacement bymeans of the two location points P of the two periodic patterns.

FIG. 21 represents an embodiment of a positioning element 10 n accordingto the invention. In this embodiment, the positioning element basicallycomprises a reference part 11 and an information-writing part 12intended to make it possible to locate the periodic pattern which isassociated with it.

The reference part 11 of each positioning element is in the form of asuccession of white and black bands, for example, so that the processingunit 4 can be used to read the part 12. This information writing part 12includes, for example, a portion 12 a for writing a row number i and aportion 12 b for writing a column number j, the two portions 12 a and 12b each being formed by five bands arranged in alignment with the whiteand black bands of the reference part 11.

In this example, each positioning element 10 n makes it possible toencode 10 bits of information (5 bits for the rows and 5 bits for thecolumns), thus making it possible to work with matrices of 32×32periodic patterns 8 n.

It can thus be understood that the use of a matrix of periodic patternsmakes it possible to increase the distance measurements up to a distancewhich is fixed only by the size of the matrix itself, and no longer bythe size of the periodic pattern considered on its own.

The bands forming the two portions 12 a and 12 b are also obtained whenetching the test marker, and black or white bands may be formedaccording to the position assigned to each positioning element.

As an example, FIG. 22 represents a positioning element 10 n which isobtained by photolithography and which is intended to precisely locate aperiodic pattern in the matrix of patterns.

The information writing part of this element is read by the processingunit from the top down, for example, and makes it possible to obtain thefollowing information by binary reading:

-   for the row=01010, which corresponds to row i=10 and-   for the column=11010, which corresponds to column j=26.

Use of the matrix of periodic patterns 8 n which are associated withpositioning elements furthermore offers the opportunity to detect aperiodic pattern lying close to the centre of the image of the sensor,which thus makes it possible to reduce the distortions due to the opticsof the objective.

According to an alternative embodiment of the invention, which isrepresented in FIG. 23, the object 5 on which the test marker 8 isplaced is intended to move both in the plane (XOY) and in the Zdirection, the displacements in the Z direction also needing to bemeasured by the method described above.

To this end, the fixed observation system 1 comprises a first matricialimage sensor 2 as well as a second matricial image sensor 21, both ofwhich are substantially contained in the plane (YOZ) which isperpendicular to the plane (XOY), that is to say perpendicular to theplane defined by the two dimensions of the periodic pattern of the testmarker 8.

Furthermore, the first sensor 2 has a sighting axis 2 a which extendsalong the axis (OZ) and the second sensor 21 has a sighting axis 21 awhich makes an angle α with the axis (OZ), this angle α being determinedwhen assembling the two sensors 2 and 21.

The two sensors are also arranged so that the point of intersection ofthe two sighting axes 2 a and 21 a lies in the vicinity of the testmarker 8.

Using this device, it is now possible to record an image of the sameperiodic pattern of the test marker 8 for each of the sensors 2 and 21.It is then sufficient to calculate the first cartesian position (x, y)of the point of intersection P, as obtained using the first sensor 2,and also to calculate the second cartesian position (x, y′) of the samepoint of intersection P as obtained using the second sensor 21. Afterthese calculations, and if the two sensors 2 and 21 are actuallycontained in the same plane (YOZ), then the cartesian values (x, y) and(x, y′) of the point of intersection P should have the same value x.

The value y′ given from the image which is obtained by the second sensor21, however, is different than the value y obtained from the image ofthe first sensor 2. This is because this value y′ depends on the valueof the angle α as well as on the position of the point of intersection Palong the Z axis.

More specifically, and after simple trigonometric operations, the valuey′ can be expressed in the following way:y′=y cos α−Z sin α

From which the value of Z is deduced, which is written in the followingway:Z=(y cos α−y′)/sin α

Simply by having the cartesian positions of the point of intersection Pfrom the two sensors 2 and 21, and from the angle α, it is thus possibleto calculate the position of the point of intersection along the Z axis.

According to this alternative embodiment, after having recorded imagesfollowing a displacement of the object 5, it is thus possible tocalculate the displacement of this object 5 along the axes X, Y and Zwith a subpixel accuracy.

According to an alternative embodiment of the device, which isrepresented in FIG. 25, the sensor 2 may also have a sighting axis 2 awhich makes an angle α2 with the axis (OZ), the sensor 2 remainingsubstantially contained in the plane (YOZ) and the sensor 21 alsoremaining in a position in which its sighting axis 21 a makes an angleα1 with the axis (OZ).

It is then sufficient to calculate the cartesian position (x, y1) of thepoint of intersection P, as obtained using the sensor 21, and also tocalculate the second cartesian position (x, y2) of the same point ofintersection P as obtained using the sensor 2. After these calculations,and if the two sensors 2 and 21 are actually contained in the same plane(YOZ), then the cartesian values (x, y1) and (x, y2) of the pointof-intersection P should have the same value x.

The value y1 given from the image which is obtained by the camera 21,however, is different than the value y2 obtained from the image of thecamera 2, these two values y1 and y2 themselves being different than thereal value y of the point of intersection P.

In this alternative embodiment as represented in FIG. 25, two similarequations are thus obtained for the sensors 2 and 21, that is to say:y 1=y.cos α1−z.sin α1 andy 2=y.cos α2−z.sin α2

z is then given by the following equation:z=(y 2.cos α1−y 1.cos α2)/(sin α1.cos α2−sin α2. cos α1)

and y is given by one or other of the following equations:y=(y 1+z.sin α)/cos α1y=(y 2+z.sin α)/cos α2

FIG. 24 represents another alternative embodiment of the device forcarrying out the method of the invention.

In this alternative embodiment, the sighting axis 2 a of the sensor 2 isarranged perpendicular with the plane (XOY) containing the periodicpattern of the test marker 8. The sensor 21 in turn has a sighting axis21 a which is perpendicular to the sighting axis 2 a of the sensor 2,and which is consequently parallel to the plane (XOY) containing theperiodic pattern of the test marker 8. A beam splitter object which isattached to the test marker 8, and which may be in the form of a cube 15or a splitter plate, is furthermore interposed between the periodicpattern 8 a or the periodic patterns 8 n of the test marker 8 and thesensors 2 and 21. In this alternative embodiment, it is possible toilluminate the test marker 8 by retro lighting in order to allow some ofthe light beam passing through the periodic pattern of this test marker8 to go in the direction of the sensor 2, while another part of thelight beam is directed toward the sensor 21. In this case, it will beunderstood that after processing by the processing unit 4, the image ofthe first sensor 2 makes it possible to determine the cartesian position(x, y) of the point of intersection P, while the image obtained from thesecond sensor 21 makes it possible to calculate the cartesian position(x, z) of the point of intersection P.

According to this alternative embodiment, these coordinates (x, y, z)are thus obtained for each position of the test marker 8.

According to another alternative embodiment of the invention, which usesa device corresponding to the device represented in FIG. 3, it is alsopossible to calculate the z displacement of the test marker 8, that isto say a displacement in a direction perpendicular to the periodicpattern of the test marker 8, while having just one matricial imagesensor.

This is because, as already seen above, the calculation of the frequencyf_(o) of the periodic pattern is carried out by the processing unit whena first image is recorded.

In the event that the test marker 8 is displaced along the Z axis, thatis to say in the event that the periodic pattern 8 a approaches thesensor 2, it will be understood that the processing of a second imagewill make it possible to obtain a new frequency f_(o)′ of the periodicpattern, as seen and recorded by the sensor 2.

Furthermore, also knowing the magnification properties of the objective3, it is possible to ascertain the position Z from a calibration curveestablished beforehand. When there is a displacement of the test marker8 along the Z direction, it is thus sufficient to take the ratio of thefrequency f_(o) to the frequency f_(o)′, this ratio being a function ofZ only, in order to obtain the value of the position of the test marker8 along the z axis from the calibration curve.

1. A method for measuring the location of an object (5) observed by afixed observation system (1) connected to a processing unit (4), inorder to generate an image composed of a matrix of pixels, and saidobject (5) being provided with a test marker (8) characterized in thatthe method comprises comprising the following steps: a test marker (8)is used comprising at least one two-dimensional periodic pattern (8 a)formed by a plurality of rows and parallel columns, which aresubstantially perpendicular to the rows, the point-like elements (9)being regularly spaced along the rows and the columns, a first image ofthe pattern (8 a) is recorded, and digital processing of the first imageof the pattern (8 a) is carried out in order to generate, from saidpattern, an image containing a first grating (R1) comprising a pluralityof regularly spaced parallel first strips (T1) and an image containing asecond grating (R2) comprising a plurality of regularly spaced parallelsecond strips (T2), the second strips (T2) being substantiallyperpendicular to the first strips (T1), and for each of the first andsecond gratings, the pixel frequency (f_(o)) of this grating (R1, R2) iscalculated along a first alignment (C_(c)) of pixels which intersectsall the strips (T1, T2) of this grating (R1, R2), the pixel frequency(f_(o)) of this grating (R1, R2) is used to define an analysis functionwhich is applied to this grating (R1, R2) along the first alignment(C_(c)) of pixels the phase and the modulus which are associated withthis grating are extracted by correlation with the analysis function inorder to calculate the cartesian position of the middle of at least onestrip (T1, T2) of the grating (R1, R2) in the direction of the firstalignment (C_(c)) of pixels, the phase and the modulus which areassociated with this grating (R1, R2) are successively extracted bycorrelation with the analysis function along a plurality of pixelalignments which are parallel to the first alignment (C_(c)) of pixels,each alignment of pixels intersecting all the strips (T1, T2) of thisgrating (R1, R2) in order to independently determine the cartesianposition of each middle of said at least one strip (T1, T2) in thedirection of each corresponding alignment of pixels, a median line (D1,D2) passing substantially through all the middles of said at least onestrip (T1, T2) is calculated for each grating (R1, R2), the median line(D1) of the first grating (R1) being perpendicular to the median line(D2) of the second grating (R2), the cartesian position of the point ofintersection (P) between the two median lines (D1, D2) is calculated,and the angle (θ) defined by the median line (D1) of the first grating(R1) and a predetermined alignment of pixels is calculated.
 2. Themethod as claimed in claim 1, in which a second image of said at leastone periodic pattern (8 a) is recorded after a displacement of theobject (5) in the space observed by the fixed observation system (1),and the cartesian position of the point of intersection (P) of the twomedian lines (D1, D2) of the first and second gratings (R1, R2) asobtained from the second recorded image is calculated in order tocalculate the displacement of the object (5).
 3. The method as claimedin one or other of claims 1 and 2, in which the digital processing ofthe first image of said at least one periodic pattern (8 a) comprisesthe following steps: a forward Fourier transform is applied to the imageof the pattern (8 a) in order to obtain the Fourier spectrum of theimage of said periodic pattern (8 a), based on the Fourier spectrum, twoindependent filtering operations are carried out in order to obtain, onthe one hand, a first filtered Fourier spectrum associated with thedirection of the columns of the periodic pattern (8 a) and, on the otherhand, a second filtered Fourier spectrum associated with the directionof the rows of the periodic pattern, and an inverse Fourier transform isapplied to each of the first and second filtered Fourier spectra inorder to obtain the image of the first grating (R1) and the image of thesecond grating (R2)
 4. The method as claimed in any one of the precedingclaims 1, in which the test marker (8) comprises a matrix of identicalperiodic patterns (8 n) arranged in parallel rows and parallel columns,which are substantially perpendicular to the rows, the periodic patterns(8 n) being regularly spaced along the rows and the columns, and eachperiodic pattern (8 n) being associated with a positioning element (10n) for locating the periodic pattern (8 n) which is associated with itinside the matrix of periodic patterns (8 n).
 5. The method as claimedin claim 4, in which each positioning element (10 n) comprises a rownumber index (i) and a column number index (j) for making it possible tolocate the pattern (8 n) which is associated with it inside the matrixof periodic patterns (8 n).
 6. The method as claimed in claim 5, inwhich the image of each row number index (i) and column number index (j)in the matrix of pixels is in the form of a barcode (12 a, 12 b) whichis read by the processing unit.
 7. The method as claimed in any one ofthe preceding claims 1, in which the fixed observation system (1)comprises a first and a second matricial image sensor (2, 21) which arecontained substantially in a first plane (yoz) perpendicular to a secondplane (xoy) defined by the two dimensions of the periodic pattern (8 a)of the test marker (8), the first and second image sensors (2, 21)having sighting axes (2 a, 21 a) each of which delimits a predeterminedangle (α1, α2) with the axis (oz) perpendicular to the second plane(xoy), and an image of said at least one periodic pattern (8 a) isrecorded by each sensor (2, 21), the first cartesian position of thepoint of intersection (P) as obtained from the first sensor (2) iscalculated, the second cartesian position of the point of intersection(P) as obtained from the second sensor (21) is calculated, and based onthe first and second cartesian positions of the point of intersection(P) and the predetermined angles (α1 , α2), the position of the point ofintersection (P) is calculated along a direction parallel to the secondplane (xoy) defined by the two dimensions of said at least one periodicpattern (8 a) and a direction (Z) perpendicular to the second plane(xoy) defined by the two dimensions of said at least one periodicpattern (8 a).
 8. The method as claimed in any one of claims 1 to 6, inwhich the fixed observation system comprises a first matricial imagesensor (2) that has a sighting axis (2 a) perpendicular to the a secondplane (xoy) defined by the two dimensions of the periodic pattern (8 a)of the test marker and a second matricial image sensor (21) that has asighting axis (21 a) parallel to the second plane (xoy) defined by thetwo dimensions of the periodic pattern (8 a) of the test marker (8)light-beam splitting object (15) being furthermore interposed betweenthe periodic pattern (8 a) and the first and second sensors (2, 21), animage of said at least one periodic pattern (8 a) is recorded by eachsensor (2, 21), and the cartesian position (X, Y) of the point ofintersection (P) in a plane parallel to the second plane (xoy) definedby the two dimensions of the periodic pattern (8 a) is calculated fromthe image obtained by the first sensor (2), and the cartesian position(X, Z) of the point of intersection (P) in a plane (XOZ) perpendicularto the first plane (XOY) defined by the two dimensions of the periodicpattern (8 a) is calculated from the image obtained by the second sensor(21).
 9. The method as claimed in any one of claims 1 to 6, in which thefrequency (f_(o)) of the periodic pattern (8 a), as calculated by theprocessing unit (4), is compared with the real frequency (F_(o)) of theperiodic pattern (8 a) in order to determine the position of the pointof intersection (P) in a direction (Z) perpendicular to the a plane(XOY) defined by the two dimensions of said at least one periodicpattern (8 a), as a function of the magnification index of the fixedobservation system (I).